有限玻色体系的严格正则系综理论研究
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摘要
囚禁外场中的冷原子的玻色-爱因斯坦凝聚(BEC)实验和其它一些介观系统如原子核、分子、原子团簇和聚合物等实验观测,是小系统的“相变”研究的原形和平台。这些实验的实现和观测激起了人们对有限体系的临界现象的理论研究新的兴趣。特别是,BEC作为量子统计相变具有本身的物理意义和研究价值,它是可操控的凝聚态物理的试验平台。实验中的粒子数几乎确定并且有限,因此实际情况与传统的采用热力学极限处理的方法所遇情况完全不同。
热力学极限下玻色气体的物理量诸如比热、凝聚比等在临界点出现锋锐的尖点或者不连续,但有限玻色体系的物理量在临界温度附近都不同程度地为圆滑曲线。虽然“不连续”相变只存在于热力学极限下,但是有限体系正如实验观测的会出现相变的先兆。本文主要是采用正则系综研究粒子数有限的理想和弱相互作用气体的热力学和统计性质,特别是在临界点邻域的性质。
在第二章,我们分别指出了巨正则系综和正则系综讨论有限玻色体系的优势和缺陷,研究了囚禁于谐振子势的任意有限粒子数的理想玻色气体的热力学行为。通过考虑体系的总的粒子数N守恒和采用鞍点近似的方法,我们得出了处在任意态平均粒子占有数的解析表达式。本章做出了囚禁势中有限粒子数的玻色气体的化学势、比热和凝聚比与温度的关系曲线图并对其进行了分析。将本章的结果与采用传统的巨正则系综方法得到的结果一一对应比较发现,它们之间的区别在低温时明显,特别系统的粒子数较小时区别尤其明显。
第三章用严格正则系综理论分别讨论了处在方盒子或者谐振子势中的粒子数有限的理想和弱相互作用玻色气体的热力学性质。当理想玻色气体处在周期或者狄利克雷(Dirichlet)边界条件的方盒子中时,我们利用严格的正则配分函数的递推关系式数值求出配分函数后,计算了一些物理量诸如化学势、比热、凝聚比、基态粒子数的均方根涨落和转变温度等,并对它们在不同的势阱下的值一一进行了比较。通过三种不同的转变温度定义,发现不同的定义得到的转变温度值的区别明显。但是,同一种定义下的转变温度值在狄利克雷边界条件比在周期性边界条件下要高,这就表明有限性效应在狄利克雷边界条件比在周期边界条件更明显。类似于理想气体的配分函数的递推关系的推导方法,我们利用博戈留玻夫(Bogliubov)理论首次推导了描述弱相互作用气体的正则配分函数递推关系式。基于正则系综配分函数的递推关系式,数值分析了不同粒子数和不同相互作用强度的玻色体系的凝聚比、比热和温度之间的关系。通过两种不同的转变温度的定义,我们分析了原子间相互作用、粒子数多少对转变温度值的影响。最后,分析了凝聚比的有限尺寸标度行为。结果表明,不同粒子数的体系的凝聚比遵从同一普适函数。
第四章讨论了理想和弱相互作用玻色气体的临界行为。对于理想玻色气体,引入依赖于具体势阱的势阱指数θ后我们讨论了处在方盒子和谐振子势中的玻色气体的凝聚比和比热的势阱尺寸标度行为。当玻色气体处在周期或者狄利克雷边界条件的方盒子时,势阱指数θ→1,我们发现表征临界行为的标度函数是普适的但与边界有关。当玻色气体囚禁在谐振子势中时,发现粒子数大小不同的体系的比热和凝聚比分别遵从各自的普适函数,并且获得θ(?)0.157。对于弱相互作用玻色气体,我们采用正则系综理论分析了比热在临界点领域的有限尺寸标度行为,并且数值获得了比热和关联长度指数的值,它们分别和实验所测得数据和以前的理论分析值一致。当体系的相互作用强弱和粒子数密度确定时,粒子数和温度大小不同的比热遵从同一形式。根据标度理论,我们讨论了弱相互作用气体的临界温度Tc的相对于热力学极限下的理想玻色气体的临界温度Tc0的位移△Tc,并且得到(?)其中b=0.42±0.05。我们将本章的结果与方格子中的4He的实验数据以及理论值进行了比较和分析。
第五章我们首先简单回顾了为何只有热力学极限下才会发生相变以及相变的分类方法,接着介绍了基于正则系综理论的有限体系的“相变”的分类方法。通过拓展刻画有限体系的复温度平面内的相变分类方法,将粒子数有限的处在不同边界条件的方盒子中的理想和弱相互作用玻色气体的相变进行了分类。结果表明,理想玻色气体的BEC在周期边界条件下为二级相变；当理想玻色气体处在狄利克雷边界条件时经历一级相变。对于处在周期性边界条件的方盒子的弱相互作用玻色气体,我们讨论了粒子数多少和原子间的相互作用强度对相变本质的影响。我们发现,有限粒子数的均匀弱相互作用的BEC为二级相变,这和无穷大系统的普适类性质一致。最后,我们讨论了粒子数大小和原子间弱相互作用的强度对有限玻色体系的转变温度的影响。
第六章列出了有限体的量子统计理论需要进一步深入研究的课题。本论文的研究结果有助于深入了解有限量子体的各种统计性质与临界行为,也为开展与此相关的实验工作(包括费米有限体、团簇、等离子体和可操控的凝聚态物质)提供了有益的理论依据。
Experiments of Bose-Einstein condensation (BEC) in ultracold trapped atomic gases or experimental observations of other mesoscopic systems such as atomic nu-clei, molecules, atomic clusters, and finite polymers, are prototypes of physical inves-tigations of transitions in small systems and have created a renewed interest in critical phenomenon for finite systems. In particular, BEC as a purely quantum-statistical phase transition has opened far-reaching prospects and has become a test laboratory where the atoms can be well manipulated in the modern condensed matter physics. The number of particles in the experiments are roughly fixed and finite, and thus the experimental situation is quite different from the traditional treatments, in which the thermodynamic limit was used.
In the finite Bose systems the physical quantities, such as specific heat and conden-sation fraction show a more or less smooth humps extending over some finite temper-ature ranges. This is quite different from the thermodynamic limit which exist a sharp peak or a discontinuity in these physical quantities. Although phase transitions occurs only in the thermodynamic limit, precursors of phase transitions in finite systems far away from the thermodynamic limit do exist, as confirmed by the experiments. Re-search topics in the present dissertation motivated by experiments of finite systems are, within a canonical ensemble treatment, aimed to find finite-size effects on the thermo-dynamics and statistical properties for ideal and weakly interacting Bose gases, espe-cially when these finite Bose systems near the transition regions.
In Chapter 2, we discuss briefly the merits as well as flaws of grand canonical and canonical ensembles, respectively. We investigate the thermodynamic behavior of ideal Bose gases with an arbitrary number of particles confined in a harmonic poten-tial. By taking into account the conservation of total number N of particles and using a saddle-point approximation, we derive the simple explicit expression of mean occu-pation number at any state of the finite system, and then numerically obtain the tem-perature dependence of the chemical potential, the specific heat, and the condensate fraction for the trapped gases with a finite number of particles. Comparing the results with corresponding those from the traditional grand canonical treatment, we find that the considerable difference between them show up for at sufficiently low temperatures, specially for the relative small numbers of Bose particles.
Within the exact canonical ensemble treatment, in Chapter 3 we investigate the ther-modynamics and finite size scaling for finite ideal and weakly interacting Bose gas in a cubic box or in a harmonic trap. Both in the box trap imposed by either periodic or Dirichlet boundary conditions (BCs) and in the harmonic trap, for an ideal gas we cal-culate several physical quantities such as the chemical potential, the specific heat, the condensate fraction, the root-mean-square fluctuations of the condensate, and the tran-sition temperature etc., and compare these quantities under different traps. We discuss the particle-number dependence of the transition by proposing several transition tem-perature definitions, where the differences among these values are considerable for the finite systems. For a weakly interacting Bose in a box with periodic boundary condi-tions, the thermodynamic properties are investigated theoretically, based on a recursion relation for the canonical ensemble partition function. In a similar manner to the case of an ideal Bose gas, we first establish the recursive scheme of the interacting finite system at finite temperatures in the framework of the Bogoliubov theory. The tempera-ture dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and non-monotonous behaviors. We discuss the finite-size scaling of conden-sate fraction at the transition temperature for the systems, showing that the calculated finite-size scaling is universal and thus independent on the various system sizes and transition temperatures.
Chapter 4 is devoted to study of critical behaviors for an ideal and weakly interact-ing Bose gas. For an ideal Bose gas, we study the trap-size scaling behaviors of the condensate fraction and the specific heat for the three different traps, introducing a trap exponentθin dependence of the trapping potential. In the box trap with periodic and Dirichlet BCs, whereθ→1, we find that the scaling functions governing the various critical behaviors are universal but respective of the BCs. The borders of universality validity are obtained numerically. In the harmonic trap, the critical behavior of the sys-tem is also found to be universal, and the trap exponent is obtained asθ(?) 0.157. For a weakly interacting Bose system, within the exact canonical ensemble treatment we study the finite-size scaling behavior of the specific heat near the critical region, and obtain the specific heat exponent and correlation length exponent which agree well with experimental data and previous theoretical predictions. For fixed interaction parameters nd particle number density, we report for the first time that, the dimensionless specific heats per particle of various system sizes and temperatures collapse onto a single form. From the scaling arguments, we study the interaction-induced shift of the transition temperature (△Tc= Tc-Tc0) and find that△Tc/Tc0= ban(?) with b= 0.42±0.05, where Tc0 is the ideal-gas thermodynamic critical temperature. We also compare our findings with experimental and theoretical results for 4He in cubic lattices, showing the different behaviors for the two systems.
In chapter 5, based on the exact canonical ensemble treatment, we generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane, and present the classifications of phase transitions in ideal and weakly interact-ing Bose gases of a finite number of particles, confined in a cubic box with different boundary conditions. In extending the classification parameters to all regions, we pre-dict that for the finite system the phase transition for periodic boundary conditions is of second order, while transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the ef-fects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on the inter-action strengths and particle numbers.
The research topics about quantum statistical theory, which deserve a deeper study for the finite quantum systems, are listed in the Chapter 6.
The results in this dissertation allow to understand well the statistical properties and critical behaviors for the finite quantum systems, and thus provide the theoretical guidelines for the future experiments including finite Fermi systems, atomic clusters, neutron starts, and manipulated condensed matter physics.
热力学极限下玻色气体的物理量诸如比热、凝聚比等在临界点出现锋锐的尖点或者不连续,但有限玻色体系的物理量在临界温度附近都不同程度地为圆滑曲线。虽然“不连续”相变只存在于热力学极限下,但是有限体系正如实验观测的会出现相变的先兆。本文主要是采用正则系综研究粒子数有限的理想和弱相互作用气体的热力学和统计性质,特别是在临界点邻域的性质。
在第二章,我们分别指出了巨正则系综和正则系综讨论有限玻色体系的优势和缺陷,研究了囚禁于谐振子势的任意有限粒子数的理想玻色气体的热力学行为。通过考虑体系的总的粒子数N守恒和采用鞍点近似的方法,我们得出了处在任意态平均粒子占有数的解析表达式。本章做出了囚禁势中有限粒子数的玻色气体的化学势、比热和凝聚比与温度的关系曲线图并对其进行了分析。将本章的结果与采用传统的巨正则系综方法得到的结果一一对应比较发现,它们之间的区别在低温时明显,特别系统的粒子数较小时区别尤其明显。
第三章用严格正则系综理论分别讨论了处在方盒子或者谐振子势中的粒子数有限的理想和弱相互作用玻色气体的热力学性质。当理想玻色气体处在周期或者狄利克雷(Dirichlet)边界条件的方盒子中时,我们利用严格的正则配分函数的递推关系式数值求出配分函数后,计算了一些物理量诸如化学势、比热、凝聚比、基态粒子数的均方根涨落和转变温度等,并对它们在不同的势阱下的值一一进行了比较。通过三种不同的转变温度定义,发现不同的定义得到的转变温度值的区别明显。但是,同一种定义下的转变温度值在狄利克雷边界条件比在周期性边界条件下要高,这就表明有限性效应在狄利克雷边界条件比在周期边界条件更明显。类似于理想气体的配分函数的递推关系的推导方法,我们利用博戈留玻夫(Bogliubov)理论首次推导了描述弱相互作用气体的正则配分函数递推关系式。基于正则系综配分函数的递推关系式,数值分析了不同粒子数和不同相互作用强度的玻色体系的凝聚比、比热和温度之间的关系。通过两种不同的转变温度的定义,我们分析了原子间相互作用、粒子数多少对转变温度值的影响。最后,分析了凝聚比的有限尺寸标度行为。结果表明,不同粒子数的体系的凝聚比遵从同一普适函数。
第四章讨论了理想和弱相互作用玻色气体的临界行为。对于理想玻色气体,引入依赖于具体势阱的势阱指数θ后我们讨论了处在方盒子和谐振子势中的玻色气体的凝聚比和比热的势阱尺寸标度行为。当玻色气体处在周期或者狄利克雷边界条件的方盒子时,势阱指数θ→1,我们发现表征临界行为的标度函数是普适的但与边界有关。当玻色气体囚禁在谐振子势中时,发现粒子数大小不同的体系的比热和凝聚比分别遵从各自的普适函数,并且获得θ(?)0.157。对于弱相互作用玻色气体,我们采用正则系综理论分析了比热在临界点领域的有限尺寸标度行为,并且数值获得了比热和关联长度指数的值,它们分别和实验所测得数据和以前的理论分析值一致。当体系的相互作用强弱和粒子数密度确定时,粒子数和温度大小不同的比热遵从同一形式。根据标度理论,我们讨论了弱相互作用气体的临界温度Tc的相对于热力学极限下的理想玻色气体的临界温度Tc0的位移△Tc,并且得到(?)其中b=0.42±0.05。我们将本章的结果与方格子中的4He的实验数据以及理论值进行了比较和分析。
第五章我们首先简单回顾了为何只有热力学极限下才会发生相变以及相变的分类方法,接着介绍了基于正则系综理论的有限体系的“相变”的分类方法。通过拓展刻画有限体系的复温度平面内的相变分类方法,将粒子数有限的处在不同边界条件的方盒子中的理想和弱相互作用玻色气体的相变进行了分类。结果表明,理想玻色气体的BEC在周期边界条件下为二级相变；当理想玻色气体处在狄利克雷边界条件时经历一级相变。对于处在周期性边界条件的方盒子的弱相互作用玻色气体,我们讨论了粒子数多少和原子间的相互作用强度对相变本质的影响。我们发现,有限粒子数的均匀弱相互作用的BEC为二级相变,这和无穷大系统的普适类性质一致。最后,我们讨论了粒子数大小和原子间弱相互作用的强度对有限玻色体系的转变温度的影响。
第六章列出了有限体的量子统计理论需要进一步深入研究的课题。本论文的研究结果有助于深入了解有限量子体的各种统计性质与临界行为,也为开展与此相关的实验工作(包括费米有限体、团簇、等离子体和可操控的凝聚态物质)提供了有益的理论依据。
Experiments of Bose-Einstein condensation (BEC) in ultracold trapped atomic gases or experimental observations of other mesoscopic systems such as atomic nu-clei, molecules, atomic clusters, and finite polymers, are prototypes of physical inves-tigations of transitions in small systems and have created a renewed interest in critical phenomenon for finite systems. In particular, BEC as a purely quantum-statistical phase transition has opened far-reaching prospects and has become a test laboratory where the atoms can be well manipulated in the modern condensed matter physics. The number of particles in the experiments are roughly fixed and finite, and thus the experimental situation is quite different from the traditional treatments, in which the thermodynamic limit was used.
In the finite Bose systems the physical quantities, such as specific heat and conden-sation fraction show a more or less smooth humps extending over some finite temper-ature ranges. This is quite different from the thermodynamic limit which exist a sharp peak or a discontinuity in these physical quantities. Although phase transitions occurs only in the thermodynamic limit, precursors of phase transitions in finite systems far away from the thermodynamic limit do exist, as confirmed by the experiments. Re-search topics in the present dissertation motivated by experiments of finite systems are, within a canonical ensemble treatment, aimed to find finite-size effects on the thermo-dynamics and statistical properties for ideal and weakly interacting Bose gases, espe-cially when these finite Bose systems near the transition regions.
In Chapter 2, we discuss briefly the merits as well as flaws of grand canonical and canonical ensembles, respectively. We investigate the thermodynamic behavior of ideal Bose gases with an arbitrary number of particles confined in a harmonic poten-tial. By taking into account the conservation of total number N of particles and using a saddle-point approximation, we derive the simple explicit expression of mean occu-pation number at any state of the finite system, and then numerically obtain the tem-perature dependence of the chemical potential, the specific heat, and the condensate fraction for the trapped gases with a finite number of particles. Comparing the results with corresponding those from the traditional grand canonical treatment, we find that the considerable difference between them show up for at sufficiently low temperatures, specially for the relative small numbers of Bose particles.
Within the exact canonical ensemble treatment, in Chapter 3 we investigate the ther-modynamics and finite size scaling for finite ideal and weakly interacting Bose gas in a cubic box or in a harmonic trap. Both in the box trap imposed by either periodic or Dirichlet boundary conditions (BCs) and in the harmonic trap, for an ideal gas we cal-culate several physical quantities such as the chemical potential, the specific heat, the condensate fraction, the root-mean-square fluctuations of the condensate, and the tran-sition temperature etc., and compare these quantities under different traps. We discuss the particle-number dependence of the transition by proposing several transition tem-perature definitions, where the differences among these values are considerable for the finite systems. For a weakly interacting Bose in a box with periodic boundary condi-tions, the thermodynamic properties are investigated theoretically, based on a recursion relation for the canonical ensemble partition function. In a similar manner to the case of an ideal Bose gas, we first establish the recursive scheme of the interacting finite system at finite temperatures in the framework of the Bogoliubov theory. The tempera-ture dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and non-monotonous behaviors. We discuss the finite-size scaling of conden-sate fraction at the transition temperature for the systems, showing that the calculated finite-size scaling is universal and thus independent on the various system sizes and transition temperatures.
Chapter 4 is devoted to study of critical behaviors for an ideal and weakly interact-ing Bose gas. For an ideal Bose gas, we study the trap-size scaling behaviors of the condensate fraction and the specific heat for the three different traps, introducing a trap exponentθin dependence of the trapping potential. In the box trap with periodic and Dirichlet BCs, whereθ→1, we find that the scaling functions governing the various critical behaviors are universal but respective of the BCs. The borders of universality validity are obtained numerically. In the harmonic trap, the critical behavior of the sys-tem is also found to be universal, and the trap exponent is obtained asθ(?) 0.157. For a weakly interacting Bose system, within the exact canonical ensemble treatment we study the finite-size scaling behavior of the specific heat near the critical region, and obtain the specific heat exponent and correlation length exponent which agree well with experimental data and previous theoretical predictions. For fixed interaction parameters nd particle number density, we report for the first time that, the dimensionless specific heats per particle of various system sizes and temperatures collapse onto a single form. From the scaling arguments, we study the interaction-induced shift of the transition temperature (△Tc= Tc-Tc0) and find that△Tc/Tc0= ban(?) with b= 0.42±0.05, where Tc0 is the ideal-gas thermodynamic critical temperature. We also compare our findings with experimental and theoretical results for 4He in cubic lattices, showing the different behaviors for the two systems.
In chapter 5, based on the exact canonical ensemble treatment, we generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane, and present the classifications of phase transitions in ideal and weakly interact-ing Bose gases of a finite number of particles, confined in a cubic box with different boundary conditions. In extending the classification parameters to all regions, we pre-dict that for the finite system the phase transition for periodic boundary conditions is of second order, while transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the ef-fects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on the inter-action strengths and particle numbers.
The research topics about quantum statistical theory, which deserve a deeper study for the finite quantum systems, are listed in the Chapter 6.
The results in this dissertation allow to understand well the statistical properties and critical behaviors for the finite quantum systems, and thus provide the theoretical guidelines for the future experiments including finite Fermi systems, atomic clusters, neutron starts, and manipulated condensed matter physics.
引文
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